Note that \(h_Y^{\rm end}\)  is by construction gauge-independent.
Here and in the following, the label “end” will refer to the end of hypermagnetogenesis.
In order to compute the time integral in Eq. (REF ), we make use of the fact that \(\left<\mathbf {E}\cdot \mathbf {B}\right>\)  is expected to reach an attractor solution, if the gauge-field production parameter \(\xi \)  and the Hubble rate \(H\)  do not vary too fast during inflation.\(3\endcsname \)For strong axion–vector coupling, the backreaction of the gauge fields onto the axion dynamics can lead to oscillations in the axion velocity and consequently in the instability parameter \(\xi \)  [1]}, [2]}, [3]}.
These oscillations are, however, damped once fermion production is included, which limits the efficiency of gauge-field production and hence reduces the effect on the axion motion [2]}.

In fact, for constant values of \(\xi \)  and \(H\) , one can show that the time dependence of \(\left<\mathbf {E}\cdot \mathbf {B}\right>\)  becomes trivial, \(\left<\mathbf {E}\cdot \mathbf {B}\right> \propto a^4\)  (see, e.g., Ref. [5]} for an explicit derivation).
This indicates that, as expected for a stationary attractor solution, the product of the electric and the magnetic field remains constant in the physical frame.
The attractor solution for \(\left<\mathbf {E}\cdot \mathbf {B}\right>\)  is moreover homogeneous and isotropic across the volume \(\mathbb {V} \gg H^{-3}\) , which allows us to drop the spatial integral in Eq. (REF ),
\(h_Y^{\rm end} = -\frac{2}{a_{\rm end}^3} \int _{-\infty }^{\tau _{\rm end}} d\tau \:\left<\mathbf {E}\cdot \mathbf {B}\right> \,.\) 
